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arXiv · 2609.15500

Uniform bounds for bubbles of neutral quadratic polynomials

Dzmitry Dudko·Willie Rush Lim
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Abstract

Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the $α'$-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.

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Dzmitry Dudko, Willie Rush Lim. 2026-09-14. Uniform bounds for bubbles of neutral quadratic polynomials. https://arxiv.org/abs/2609.15500

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